Let \(\mathcal V\) be a symmetric monoidal \(\infty\)-category, \(C\) an \(\mathbb E_2\)-algebra (or merely an \(\mathbb A_2 \otimes \mathbb E_1\)-algebra) and \(A\xrightarrow{f} B \xrightarrow{g}C\) be maps of \(\mathbb E_1\)-algebras. We define the space \(\mathrm{PreBraid}_{\mathcal V}(f)_{/C}\) of prebraidings on \(f\) over \(C\) to be the space \(\mathbb T^{\mathrm{Alg}_{\mathbb E_1}(\mathcal V)}_2(f)_{/C}\) from definition 7.3.1.
Unpacked, a prebraiding on \(f\) over \(C\) is therefore a prebraiding on \(f\) together with an identification of the induced prebraiding on \(g\circ f\) with the one induced by the \(\mathbb E_2\)-structure of \(C\).